--- a/doc/kishna.tex Mon Dec 08 23:47:48 2008 +0000
+++ b/doc/kishna.tex Mon Dec 08 23:49:48 2008 +0000
@@ -1,4 +1,4 @@
-\documentclass[a4papre,12pt]{article}
+\documentclass[a4paper,12pt]{article}
\usepackage[utf8]{inputenc}
\usepackage[english]{babel}
@@ -163,26 +163,26 @@
The mathematical formulation of the Runge-Kutta method:
If we have an initial value problem of the form
-\begin{equation}
+\begin{equation*}
y' = f(t, y), \quad y(t_0) = y_0.
-\end{equation}
+\end{equation*}
The we can describe the RK4 method for this problem by equations
-\begin{align}
+\begin{align*}
y_{n+1} &= y_n + \tfrac{1}{6}h\left(k_1 + 2k_2 + 2k_3 + k_4 \right) \\
t_{n+1} &= t_n + h \\
-\end{align}
+\end{align*}
where $y_{n+1}$ is the RK4 approximation of $y(t_{n+1})$, and
-\begin{align}
+\begin{align*}
k_1 &= f(t_n, y_n) \\
k_2 &= f(t_n + \tfrac{1}{2}h, y_n + \tfrac{1}{2}h k_1) \\
k_3 &= f(t_n + \tfrac{1}{2}h, y_n + \tfrac{1}{2}h k_2) \\
k_4 &= f(t_n + h, y_n + h k_3) \\
-\end{align}
+\end{align*}
The next value $(y_n+1)$ is determined by the present value $(y_n)$,
the product of the interval $(h)$ and an estimated slope that is
defined as $\frac{1}{6}h\left(k_1+2k_2+2k_3+k_4\right)$.
-something network related?
+% TODO: something network related?
\section{Known bugs}
A lots of them (or maybe not).